# Quickly Compute Definite Integrals Using the Fundamental Theorem

Here is a super-duper shortcut integration theorem that you’ll use for the rest of your natural born days — or at least till the end of your stint with calculus. This shortcut method is all you need for most integration word problems.

**The fundamental theorem of calculus (second version or shortcut version):** Let *F* be any antiderivative of the function *f;* then

This theorem gives you the super shortcut for computing a definite integral like

the area under the parabola *y* = *x*^{2} + 1 between 2 and 3. You can get this area by subtracting the area between 0 and 2 from the area between 0 and 3, but to do that you need to know that the particular area function sweeping out area beginning at zero,

(with a *C* value of zero).

The beauty of the shortcut theorem is that you don’t have to even use an area function like

You just find any antiderivative, *F* (*x*), of your function, and do the subtraction, *F* (*b*) – *F* (*a*). The simplest antiderivative to use is the one where *C* = 0. So here’s how you use the theorem to find the area under your parabola from 2 to 3.

is an antiderivative of *x*^{2} + 1. Then the theorem gives you:

Regardless of the function, this shortcut works, and you don’t have to worry about area functions. All you do is *F* (*b*) – *F* (*a*).

Here’s another example: What’s the area under *f* (*x*) = *e** ^{x}*, between

*x*= 3 and

*x*= 5? The derivative of

*e*

*is*

^{x}*e*

*, so*

^{x}*e*

*is an antiderivative of*

^{x}*e*

*, and thus*

^{x}What could be simpler?

**Areas ***above ***the curve and ***below ***the ***x***-axis count as ***negative ***areas.** Before going on, it’s important to touch on negative areas. Note that with the two examples shown here, the parabola, *y* = *x*^{2} + 1, and the exponential function, *y* = *e** ^{x}*, the areas you’re computing are

*under*the curves and

*above*the

*x*-axis. These areas count as ordinary,

*positive*areas. But, if a function goes below the

*x*-axis, areas above the curve and below the

*x*-axis count as

*negative*areas.

Okay, so now you’ve got the super shortcut for computing the area under a curve. And if one big shortcut wasn’t enough to make your day, This table lists some rules about definite integrals that can make your life much easier.